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C5⋊(C11⋊(D73⋊C3)) #37562813

order24090
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C11⋊(D73⋊C3))
unambiguous semidirect productC11⋊(C5⋊(D73⋊C3))
unambiguous semidirect product(C73⋊C3)⋊D55
unambiguous semidirect productD4015⋊C3

Splittings

Fitting, socle, derived, HallC5×C11×C73 ⋊ C2×C3
HallC5×C11×C73⋊C3 ⋊ C2
HallD4015 ⋊ C3
HallC11×C73⋊C3 ⋊ D5
HallC5×C73⋊C3 ⋊ D11
HallC11×C73 ⋊ C3×D5
HallC5×C73 ⋊ C3×D11
HallC73⋊C3 ⋊ D55
SylowC73 ⋊ C3×D55
HallC5×C11 ⋊ D73⋊C3
SylowC11 ⋊ C5⋊(D73⋊C3)
SylowC5 ⋊ C11⋊(D73⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C11×C73C2×C3
LCSG ▷ C5×C11×C73
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 89
transitive degreeμt(G) = 4015

A minimal faithful permutation representation — 2 generators on 89 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,60)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,48,57,34,49,63)(37,42,62,50,51,58)(40,67,65,54,55,53)(52,61,68,56,64,72)(59,73,71,69,70,66)(75,76)(77,78)(79,80)(81,82)(83,84)(86,87)(88,89);(1,2,5,11,19,32,33,20,4,8,13,23,37,52,54,36,10,16,24,38,53,66,69,51,18,29,39,41,57,68,73,64,31,48,55,58,60,47,62,67,49,46,61,70,72,63,28,43,44,27,42,59,71,65,45,15,26,14,25,40,56,50,35,22,7,6,12,21,34,30,17,9,3)(74,75,77,79,81,83,84,82,80,78,76)(85,86,88,89,87)

Polycyclic presentation:

2,3,5,11,73|c^a=c^4,d^a=d^10,e^a=e^72,e^b=e^8

Conjugacy classes & character table

747 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24090, 33);